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Journal of Economic Psychology 29 (2008) 792–809
Contents lists available at ScienceDirect
Journal of Economic Psychology
journal homepage: www.elsevier.com/locate/joep
Rational ignoring with unbounded cognitive capacity
Nathan Berg a,b,*, Ulrich Hoffrage c
a
Center for Adaptive Behavior and Cognition, Max Planck Institute for Human Development, Berlin, Germany
School of Economic, Political and Policy Sciences (EPPS), University of Texas at Dallas, 800 W. Campbell Road,
GR31 Richardson, TX 75083-3021, USA
c
University of Lausanne, Faculty of Business and Economics (HEC), Batiment Internef, CH-1015 Lausanne, Switzerland
b
a r t i c l e
i n f o
Article history:
Received 13 May 2006
Received in revised form 4 December 2007
Accepted 3 March 2008
Available online 25 March 2008
JEL classification:
D60
D83
D11
PsycINFO classification:
2340
4120
2630
2240
2346
2343
a b s t r a c t
In canonical decision problems with standard assumptions, we
demonstrate that inversely related payoffs and probabilities can
produce expected-payoff-maximizing decisions that are independent of payoff-relevant information. This phenomenon of rational
ignoring, where expected-payoff maximizers ignore costless and
genuinely predictive information, arises because the conditioning
effects of such signals disappear on average (i.e., under the expectations operator) even though they exert non-trivial effects on payoffs and probabilities considered in isolation (i.e., before
integrating). Thus, rational ignoring requires no decision costs, cognitive constraints, or other forms of bounded rationality. This
implies that simple decision rules relying on small subsets of the
available information can, depending on the environment in which
they are used, achieve high payoffs. Ignoring information is therefore rationalizable solely as a consequence of the shape of the stochastic payoff distribution.
Ó 2008 Elsevier B.V. All rights reserved.
Keywords:
Ignoring
Value of information
Heuristic
Bounded rationality
Ecological rationality
* Corresponding author. Address: School of Economic, Political and Policy Sciences (EPPS), University of Texas at Dallas,
800 W. Campbell Road, GR31 Richardson, TX 75083-3021, USA. Tel.: +1 972 883 2088; fax: +1 972 883 2735.
E-mail addresses: [email protected] (N. Berg), [email protected] (U. Hoffrage).
URLs: http://www.utdallas.edu/~nberg/ (N. Berg), http://www.hec.unil.ch/people/uhoffrage (U. Hoffrage).
0167-4870/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.joep.2008.03.003
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1. Introduction
Do environments exist in which unboundedly rational optimizers ignore payoff-relevant information? If so, can these environments be characterized in terms of visible markers found in real-word
decision-making environments? This paper provides an affirmative answer to the first question and
makes a new prediction that may help address the second.
The bounded rationality literature provides abundant theoretical and empirical support for the
claim that information-frugal decision rules, which rely on small subsets of information available in
the decision maker’s environment, can come close to matching the performance of information-hungry decision rules based on constrained optimization (Baucells, Carrasco, & Hogarth, in preparation;
Conlisk, 1996; Gigerenzer and Goldstein, 1996; Gigerenzer and Selten, 2001; Martignon and Hoffrage,
2002; Rodepeter and Winter, 1999; Simon, 1982). Such findings support explanations for the success
of information-frugal decision rules that emphasize savings of decision time, effort in computation,
and other cognitive resources in scarce supply.
There is another explanation, however – one that has been largely overlooked – that may account
for why successful decision rules can rely on only small subsets of the non-redundant predictors available in the environment. This new explanation focuses on the joint structure of information and payoffs (as described by the joint distribution of stochastic payoffs and observable conditioning
information) which – apart from any limitations or bounds on human cognition – is capable of
rewarding decisions that ignore objectively predictive information.1 The focus of this paper is on signals that have zero, or near-zero, value even though they are non-trivially correlated with future events
affecting payoffs. In such cases, the signal is genuinely predictive of payoff-relevant outcomes but, because of inversely related effects on probabilities and payoffs that cancel under the expectations operator, do not affect maximized expected utility, and therefore rationalize ignoring.
In pursuing mechanisms that rationalize ignoring without cognitive constraints, we do not mean to
suggest that cognitive constraints are uninteresting or play an unimportant role. Rather, our motivation stems from the observation that many, if not all, aspects of human cognition can be alternatively
viewed as enabling or limiting, depending on the specifics of the particular task environment. For
example, limitations on human memory play an important role in enabling individuals to detect broad
patterns, discover generalizations, and engage in abstract thinking (Cosmides & Tooby, 1996; Gigerenzer, 2005; Hertwig & Gigerenzer, 1999; Kareev, 2000; Schooler & Hertwig, 2005). Conversely, very
large endowments of working memory may be disabling in some settings, leading to pathologically
poor performance in task domains requiring summarization, encapsulation and compression of informational stimuli (Luria, 1968).
Together, internal cognitive constraints and the external structure of the environment are likely to
play complementary roles in explaining conditions under which informational frugality is adaptive or
beneficial. To isolate the role of the environment, however, in selecting for informationally frugal
strategies (referred to in the biology literature as coarse behavior (Bookstaber & Langsam, 1985)), this
paper begins by assuming that decision makers are unboundedly rational. In most of the models
developed below, cognitive constraints and decision-making costs play no role, and subjective beliefs
are assumed to coincide perfectly with objective frequencies. These unrealistic assumptions serve as
controls in what is intended as a thought experiment whose goal is to cleanly identify environmental
structure that can account for behavioral insensitivity to objectively predictive information. One version of the model reintroduces cognitive limitations in the form of just-noticeable differences to demonstrate their positive interaction with environmental structure that favors ignoring, which results in
significant enlargement of the rational ignoring set.
This approach is consistent with Herbert Simon’s research program on bounded rationality, which
emphasized the interplay between external environment and internal cognitive processing, rather
than the exclusive focus on errors and biases commonly identified with bounded rationality today. Simon states (1990, p. 7): ‘‘Human rational behavior is shaped by a scissors whose blades are the struc1
There is a closely related literature on the value of information (Blackwell, 1953; Delquié, 2006; Hilton, 1981; Lawrence, 1999),
where the value of a signal is defined as expected utility (conditioning on the signal) minus unconditional expected utility, with
expectations taken a second time with respect to the signal’s marginal pdf.
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ture of task environments and the computational capabilities of the actor.” Studying the roles of both
blades, Simon argues, is needed if economics and psychology are to explain how decision rules come
into existence, proliferate, and recede.
The plan of this paper is as follows. Section 2 provides background from respective economics and
psychology literatures relating to the phenomenon of ignoring information. Section 3 describes the
general decision model. Section 4 analyzes the case in which actions are chosen from a discrete set,
and Section 5 analyzes the case in which the choice set is a continuum. Section 6 concludes with a
discussion of how these results can be interpreted and their implications for bounded rationality.
2. Background
There is abundant evidence that human subjects in psychological experiments systematically
ignore information. Explanations for why patterned ignoring is an empirical regularity in human subjects go back at least to Stroop (1935), with antecedents in William James’ (1890) implicitly probabilistic interpretation of memory. The work of Gerard (1960), Walker and Bourne (1961) and Posner
(1964) established that information reduction is a critical feature of cognitive function in signal detection tasks. Later studies of information reduction and signal extraction led to theories of automaticity
(Hasher & Zacks, 1979; Kahneman & Treisman, 1984; Logan, 1980; Logan & Zbrodoff, 1998), which implied an underlying purpose for sub-maximum utilization of available information, namely, re-allocation of scarce cognitive resources to higher priority components of the task. Recent evidence (DishonBerkovits & Algom, 2000; Godijn & Theeuwes, 2002; Hutchison, 2002) confirms the existence and
manipulability of automaticity, perceptive imperviousness, and Stroop effects.
Within economics, systematic information-processing irregularities have become a priority because of links to important policy issues such as pension finance, healthcare decisions, and immigration (Dow, 1991; Cutler, Porterba, & Summers, 1989; Mullainathan, 2002). Conlisk (1996) surveys the
bounded rationality literature within economics, describing how problem-solving costs have been
incorporated into neoclassical theory. One observes in this literature that frugality is not a necessary
consequence of bounded rationality. On the contrary, constrained optimization problems with explicit
decision costs frequently lead to more complicated analytical problems and decision rules with greater informational requirements (e.g., Sargent, 1993). Lipman (1991) identifies a different problem apart
from increasing complexity in modeling bounded rationality as cognitively constrained optimization.
Given this paper’s interest in environments that reward individuals who ignore, it is relevant to
note that normative claims in economics about ignoring information have been mostly negative. Aside
from exceptions such as Carrillo and Mariotti (2000) and Aghion et al.’s (1991) work on optimal experimentation, the standard view interprets information suppression as deriving from limitations standing in the way of full optimization rather than as an adaptive response facilitating enhanced
performance. Mullainathan (2002), for example, acknowledges that his economic approach to memory does not deal at all with the possibility of beneficial or adaptive information suppression. In contrast, computer scientists (Emran & Ye, 2001; Schein, Popescul, Ungar, & Pennock, 2002) and decision
theorists (Jones & Brown, 2002) report that ignoring can be beneficial in some models of computer
memory. In other specialized applications, such as music listening (Bigand, McAdams, & Foret,
2000) and chess playing (Reingold, Charness, Schultetus, & Stampe, 2001), there is a clear association
between ignoring information and high levels of performance. Psychological work on robustness and
flexibility when moving from one environment to another also suggests the possibility of beneficial
ignoring (Ginzburg, Janson, & Ferson, 1996; Czerlinski, Gigerenzer, & Goldstein, 1999; Hertwig & Todd,
2003). Hogarth and Karelaia (2005) demonstrate less-is-more effects in simulated binary choice with
continuous cues. Extending these interesting simulation studies with formal analysis, Baucells et al.
(in preparation), and Hogarth and Karelaia (2006, 2007) characterize large classes of decision problems according to the structure of the environment in which simplifying heuristics perform well,
emphasizing the practical implication for managers – that it is often more beneficial to identify the
most important decision factors (and rank them) than to compute optimal decision weights for each.
This paper’s approach to ignoring differs from its predecessors in that virtually all the idealized
assumptions of standard neoclassical theory are present. The setup allows us to show that bounded
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rationality, distorted beliefs, and anomalous preferences are not required to rationalize ignoring. Also
not required are non-linear probability weights or other departures from expected utility theory (c.f.,
Kahneman & Tversky, 1973). In the analysis that follows, special emphasis is given to the domain-specific nature of rationality by evaluating decision procedures in terms of whether they are well matched
to the environments in which they are used, referred to elsewhere as ecological rationality (Gigerenzer
& Todd, 1999; Smith, 2003).
3. The decision problem
Let x represent a continuous or discrete random variable on the set X indexing states of nature,
which are assumed to be ex ante unobservable. The vector of cues X represents observable environmental factors freely available for formulating expectations about x. Assumption 1 rules out redundancy, or perfect collinearity, among cues:
A1 (Non-redundancy). E[XX0 ] exists and is full rank.
The trivial case of cues with no predictive power is similarly ruled out by requiring that each component of X is relevant with respect to state probabilities (but not necessarily with respect to expected
payoffs). For a cue to be state-relevant, there must be a value in its support at which the conditional
expectation of x is non-constant with respect to X:
A2 (State-relevancy). E[xjX] is non-constant in each component of X for some value on the support of X.
The decision variable labeled a (for action) takes on values in the action space, denoted A. It turns
out that the cardinality of A is itself an important parameter in the general decision problem, and the
distinction between discrete-action versus continuous-action cases is highlighted in the analysis
below.
The function fxjX(x, X, a) denotes the conditional density of x given X, which may or may not depend
on a. If, for example, the decision maker’s choice of how much to eat this year, a, directly affects the
probability of a heart attack (denoted by x = heart attack) conditional on the high blood pressure reading (X = high), then the conditional density fxjX(x, X, a) would naturally depend on a.
The payoff function p(x, X, a) serves to rank conditional distributions of x by the expected-payoff
criterion. The function p(x, X, a) may or may not depend on X. For example if the event of no heart attack yields higher payoffs conditional on a low blood pressure reading than on a high reading, then the
payoff function p(x, X, a) depends non-trivially on X. Thus, endogeneity of frequencies (i.e., fxjX dependent on a) and signal-dependent utility (i.e., p dependent on X) are possible, but not required, elements
of the set-up.
The decision-making environment is defined as any pair of conditional density and payoff functions,
denoted {fxjX(x, X, a), p(x, X, a)}. Assuming that the necessary technical requirements for existence of a
unique maximum hold, the decision problem is specified in the usual way, with constraints on action
embedded in the definition of A, and payoff-maximizing decision rules defined as solutions to the
problem:
Z
pðx; X; aÞfxjX ðx; X; aÞdX;
ð1Þ
max
a2A
X
where the integral above is a Lebesgue integral accommodating both discrete and continuous random
variables as special cases. The solution to (1) is denoted a*, which is in general a function of the full
vector of decision cues X (i.e., a* = a*(X)).
Rational ignoring arises when a*(X) is independent of at least one element in X. The goal then is to
identify conditions on {fxjX(x, X, a), p(x, X, a)} under which the link between X and a*(X) is broken, despite the dependence of E[xjX] on X. Note that the variables x, X, and a can be discrete or continuous.
Subsequent sections show that the cardinality of the supports of x and X matters very little compared
to whether a is discrete or continuous, which profoundly affects the size of the set of rational ignoring
environments. In the next two sections, we first investigate the case in which a is discrete, followed by
the case in which a is continuous.
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4. Environments where action is a discrete variable
The simplest version of the model is when the state x 2 {L, R}, the signal X 2 {0, 1}, and the action
a 2 {U, D} are all binary variables. Consider the payoff matrix below in which rows represent actions
and columns represent states of nature:
ð2Þ
:
Zero in the northeast cell is an innocent normalization. The symbols gL and gR represent the marginal payoffs of moving from suboptimal to optimal action in each state, and d represents an unavoidable state-dependent component of payoffs. Obviously, if one action is payoff dominant (which, in this
payoff matrix, requires that gL and gR have opposite signs), then states of nature are irrelevant to the
decision maker, and information that helps predict those states is worthless. To guarantee that states
are relevant, we assume (without loss of generality) that gL > 0 and gR > 0. This assumption implies
that a = U is the optimal action in state x = L, and a = D is the optimal action in state x = R:
A3 (No uniformly best action across states). Conditional on x, the optimal action is non-constant with
respect to x.
A3 implies that states of nature are non-trivially relevant when choosing actions, because the best
action will be different depending on which state of nature is realized. In the two-state, two-action
model above, assumption A3 reduces to the two inequalities gL > 0 and gR > 0.
4.1. Unconditional action
As a benchmark against which to compare the optimal action rule when conditioning on X, this section develops an expression for the unconditionally optimal action rule when X is not used. Denote the
unconditional probability of state L as:
p Prðx ¼ LÞ:
ð3Þ
The expected payoff as a function of actions is given by:
E½pðx; aÞ ¼
pðg L þ dÞ
if a ¼ U;
pd þ ð1 pÞg R
if a ¼ D:
ð4Þ
Assuming expected-payoff maximization, the unconditional action rule is
a0 ¼ U if
p
g
> R;
1 p gL
a0 ¼ L if
p
g
< R;
1 p gL
ð5Þ
p
and indeterminate in case 1p
¼ ggRL . Unconditional action depends on the difference between the odds
of state L and the ratio of marginal payoffs with respect to action.
4.2. Conditional action
Now suppose the decision maker facing payoff matrix (2) observes X, which is known to be correlated with x and is therefore useful for helping predict states. For now, assume that X has a two-element support {0, 1}. (The case of continuous signals is described in a later section.) Denote the
conditional probabilities of state L given X as
fi Prðx ¼ LjX ¼ iÞ;
i 2 f0; 1g;
such that f 1 > p > f0 :
ð6Þ
The inequalities f1 > p > f0 imply that the signal X has strictly positive correlation with states of nature,
indicating a greater chance of state L (relative to the unconditional frequency p) when X = 1, and a lower chance of L when X = 0.
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We investigate conditions under which a* is independent of X even though X objectively helps predict action-relevant states of nature in the sense of (A3). Conditional on X, the expected-payoff matrix
is as follows, where rows correspond to actions, and columns correspond to realized values of X that
determine the conditional probabilities used in computing expectations
ð7Þ
In the conditional model, the marginal expected payoff of moving from a = D to a = U is denoted D1 or
D0, depending on the observed value of X through the index i:
fi
g
R ð1 fi Þg L ; i ¼ 0; 1:
ð8Þ
Di fi g L ð1 fi Þg R ¼
1 fi g L
Thus, D1 and D0 represent marginal gains by moving from a = D to a = U, conditioning on X = 1 and
X = 0 respectively. The first term on the right-hand side, figL, is the expected gain from correctly choosing a = U in state L. The negative term that follows, (1 fi)gR, is the expected loss incurred by incorrectly choosing a = U in state R.
Using the notation above, one can write an expression for the optimal action conditional on X. The
inequalities f1 > p > f0, together with the assumption that X and x have non-zero correlation, imply
that D1 > D0. Thus, there are three distinct action functions to consider, corresponding to the three subsets of the parameter space given by the cases D1 > D0 > 0, 0 > D1 > D0, and D1 > 0 > D0. The boundary
cases D1 = 0 or D0 = 0 imply obvious indeterminacies that follow from trivial cases where it does not
matter (in expectation conditional on X) which action is chosen. Aside from these trivial boundary
cases, the three action rules (Eqs. (9)–(11) below) cover the entire universe of environments parameterized by h [gL, gR, f1, f0]
(
U for X ¼ 1;
ð9Þ
D1 > D0 > 0 ) a ðXÞ ¼
U for X ¼ 0:
(
D for X ¼ 1;
ð10Þ
0 > D1 > D0 ) a ðXÞ ¼
D for X ¼ 0:
(
U for X ¼ 1;
ð11Þ
D1 > 0 > D0 ) a ðXÞ ¼
D for X ¼ 0:
The main observation about the conditional action functions above is that, in two out of three cases, a*
is independent of X even though X helps predict payoff-relevant x. Parameter values h corresponding
to the first two cases (D1 > D0 > 0 and 0 > D1 > D0) are referred to as the rational ignoring set. It is important to recall that no best strategy across both values of x exists, because of the assumption (A3) that
gR and gL are strictly positive. Rather, state-invariant dominance of one action over the other seen in
(9) and (10) emerges only because of uncertainty and, in particular, the interaction of conditional
probabilities and the marginal gains from correct actions. In the payoff matrix (2), U is the correct action in state L, and D is the correct action in state R. This much is unambiguous. Moving to the conditional expected-payoff matrix (7), however, it becomes possible that D1 and D0 have the same sign.
This implies (in case D1 and D0 are both positive) that U is the expected-payoff maximizer regardless
of X, or (in case D1 and D0 are both negative) that D is dominant regardless of X.
4.2.1. Result
Given a state-dependent matrix of the form (2) that contains no payoff-dominant rows, and a binary decision cue X satisfying Pr[x = LjX = 1] > Pr [x = L] > Pr[x = LjX = 0], the expected-payoff-maximizing action a* is independent of X whenever D1D0 > 0, where Di = figL (1 fi)gR, i = 0, 1, which represent
marginal expected-payoffs conditional on X when moving from action D to U.
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4.3. Determinants of rational ignoring
Rational ignoring occurs in Eqs. (9) and (10), but not (11). In the two-state, two-action environment, the determinants of rational ignoring can be examined analytically by considering an ignoring
index, denoted as F, which depends on the parameter h = [gL,gR,f1,f0] given in the environment:
f1
gR
f0
gR
ð1 f1 Þð1 f0 Þg 2L :
FðhÞ D1 D0 ¼
ð12Þ
1 f1 g L
1 f0 g L
Rational ignoring if and only if F(h) > 0. To decide whether to pay attention to X, only the sign of F(h) is
required.
As one would expect, when f1 = f0 (i.e., the signal X offers no predictive benefit), then X should indeed be ignored:
FðhÞjf1 ¼f0 ¼f ¼ ½fg L þ ð1 f Þg R 2 > 0:
ð13Þ
Apart from this trivial case, there are a large number of more interesting cases where (12) is positive.
One can additionally show that the rational ignoring set covers a large subset in the universe of admissible values of h.
For illustration, we describe one parameterization in the rational ignoring set represented by rational numbers: gR/gL = 2/5, f1/(1 f1) = 2, and f0/(1 f0) = 1/2 (or, equivalently, f1 = 2/3, and f0 = 1/
3). Because the ratio of marginal benefits with respect to optimal action (gR/gL) is less than both odds
ratios, the formula in (12) is clearly positive, and it is therefore optimal ex ante to ignore the signal.
This is interesting because, as the large difference in conditional probabilities shows, the signal is quite
informative about x, and x matters significantly in terms of payoffs. Nevertheless, because the marginal benefit of correct versus mistaken action is two and one half times as great in state x = R as in
x = L, expected payoffs are maximized by choosing the action that is correct in state x = R, regardless
of which signal is observed (i.e., regardless of which state is anticipated). Thus, it pays to ignore the
signal and play it safe by being correct in state x = R and accepting being wrong when x = L, which
costs only 2/5 what it costs to be wrong when x = R.
f0
1 – f0
45 o line
Inadmissible
parameter values
Rational
ignoring
gR/gL
Rational
ignoring
gR/gL
Expected utility maximizers
pay attention to X
f1
1 – f1
Fig. 1. Rational ignoring environments in the two-state, two-action mode.
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Fig. 1 sketches the admissible range of parameter values, which is represented by the infinite triangle in the northwest quadrant between the x-axis and the 45-degree line, and the large rational
ignoring
set within it. Coordinates on the x-axis represents the odds of x = L conditional
on X = 1,
f1
f0
,
and
coordinates
on
the
y-axis
represent
the
odds
of
x
=
L
conditional
on
X
=
0,
. By defi1f1
1f0
f1
f0
nition, the inequality f1 > f0 holds, and therefore 1f1 > 1f0 holds, too, indicated in Fig. 1 by the fact that
points above the 45-degree line are inadmissible. The ignoring set as depicted in Fig. 1 consists of pairs
of odds ratios where both quantities are on the same side of the marginal benefit ratio gR/gL
f1 =ð1 f1 Þ > f0 =ð1 f0 Þ > g R =g L ;
or g R =g L > f1 =ð1 f1 Þ > f0 =ð1 f0 Þ:
ð14Þ
Expected-payoff maximizers need to pay attention to the x-correlated signal X only for odds ratios in
the rectangle along the x-axis formed by the vertical line f1/(1 f1) = gR/gL and horizontal line f0/
(1 f0) = gR/gL. Everywhere else in the admissible parameter range is part of the rational ignoring set.
4.4. Seatbelting example
A real-world decision task that can be modeled as an instance of the simple two-state, two-action
model above is whether or not to wear a seatbelt. Let L denote the state the world when a serious auto
accident capable of producing an injury occurs, and R denote the complementary state of no such accident. Then f1 denotes the probability of a serious auto accident conditional on a signal that indicates
higher than average risk, and f0 denotes the same accident probability conditional on a signal indicating lower than average risk.
Some drivers always wear seatbelts, and some never wear them. Others wear seatbelts conditionally, perhaps weighing risk factors such as time of day, day of the week, time of year, road conditions,
the driver’s level of sleepiness, and speed. The model provides an unambiguous explanation attributing this heterogeneity to variation in gR/gL, which reflects different subjective assessments of convenience, comfort and style premiums associated with not seatbelting in no-accident states of the
world and the value of avoiding injury in states of the world in which accidents do occur. Those
who experience a large convenience, comfort or style premium by not seatbelting in no-accident
states of the world have relatively large values of gR/gL. On the other hand, those whose primary goal
is to avoid injury in the event of an accident have very small values of gR/gL.
The behavioral phenomenon of habitual seatbelt usage, without weighing easily observable signals
that shift conditional risks up and down, may appear trivial upon first consideration. Further reflection, however, reveals the high stakes involved in seatbelting decisions, with life and death contingencies hanging in the balance and the possibility of weighing an abundance of easy-to-observe risk
factors.2 Although the base rate of auto accident injuries is low, factors such as time of day, month of
the year, state within the US and, most importantly, whether one is driving on a two-lane highway, condition fatality and injury probabilities by a factor of five or more.
Seatbelting decisions have been the target of intense policy interventions aimed at persuading
more drivers to buckle up. According to the US Department of Transportation (National Highway Traffic Safety Administration, 2005), seatbelts reduce the risk of serious injury in auto accidents by 50% or
more, and yet 5–35% of the population (depending on the state within the US) choose not to wear a
seatbelt.3 A simple economic model that accounts for behavioral variation would therefore seem
desirable.
According to the model, the expected-payoff-maximizing action rule stops conditioning on X (i.e.,
rational ignoring prevails) in two cases. First, if the odds of an accident is uniformly greater than ggRL
(regardless of whether X suggests high or low risk), then the driver automatically seatbelts for all real-
2
The World Health Organization reports that auto accidents cause more than 100,000 injuries and 3000 fatalities worldwide
each day (Peden et al., 2004). For evidence on signals that condition the probability of accidents, see Elvick and Vaa (2004) or the
US Department of Transportation’s National Highway Traffic Safety Administration (National Highway Traffic Safety Administration, 2005). NHTSA research papers, data server and additional links appear at http://www.nhtsa.dot.gov.
3
Some economists dispute the claim that seatbelts save lives, based on methodological critiques and alternative data from the
Center for Disease Control (Garbaez, 1990). Notwithstanding, there appears to be broad consensus among many researchers that
seatbelts are of great benefit in the event of an accident.
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izations of X and therefore stops paying attention to it. Similarly, if the odds of an accident is uniformly
less than ggR , then the driver never wears a seatbelt regardless of X. Even though the odds of an accident
L
are very small, it is worth investigating the magnitudes required for rational ignoring to be plausible,
f0
f1
> ggR or ggR > 1f
, versus conditional seatbelting, which occurs when
as indicated by the inequalities 1f
0
1
L
L
gR
f1
f0
>
>
:
1f1
gL
1f0
The NHTSA reports what are probably the most widely cited fatality and injury statistics in the US.
Based on aggregation of all police-reported motor vehicle crashes in 2004, more than 33,000 were
killed, and 2.5 million were injured. Injury rates appear quite different depending on whether one normalizes by population, registered cars, total miles driven, or the number of licensed drivers. We focus
on one widely used statistic: The unconditional annual rate of auto injuries, which was 1402 injuries
per 100,000 licensed drivers in 2004. Of course, this statistic averages over seatbelt users and nonusers.
If the annual auto injury rate is 1402/100,000, then the unconditional daily risk of injury could be
computed by dividing through by 365: 1402/36,500,000.4 Since most accidents occur under high-risk
conditions, we assume that the unconditional rate computed above is roughly equal to f1, and then compute f0 according to the assumption that the high-risk signal raises the conditional risk of injury by a factor of 5:
f1 ¼ 1402=36; 500; 000;
and
f 0 ¼ f1 =5:
ð15Þ
Given these conditional frequencies, one may then investigate the plausibility of the two rational
ignoring conditions:
gR
f1
>
¼ 0:0000384
g L 1 f1
or 0:0000077 ¼
f0
g
> R:
1 f0 g L
ð16Þ
These inequalities imply that, if the marginal benefit of not seatbelting in no-accident states of the
world (in terms of convenience, comfort and style) is 1/25,000 (i.e., 0.0000384 rounded to the nearest
one hundred thousandth) times as beneficial as seatbelting in an accident state of the world – or more,
then it is reasonable to ignore risk signals and never wear seatbelts. On the other hand, if the benefit of
not wearing a seatbelt is less than 1/125,000 (i.e., an approximation of 0.0000077) times the seatbelting benefit, then expected-payoff maximization leads to unconditional seatbelting.
Unconditional seatbelting is a case of rationally ignoring signals that indicate lower than average
risk. Even though these signals reduce conditional risk by a factor of five or more, they are not worth
paying attention to for people who place a relatively low premium on not having to wear a seatbelt
when no accident occurs. For intermediate values of gR/gL, the optimal action conditions on X, manifested in the behavior of drivers who wear seatbelts only on the highway, at night, or conditional on
some other signal of elevated risk. For all other parameter combinations, rational ignoring prevails.
4.5. Rates of rational ignoring
To complement Fig. 1’s depiction of the large size of the rational ignoring set in the discrete-actionspace case, this section reports results from a simulation whose purpose is to investigate the frequency with which the rational ignoring condition, F(h) > 0, holds. Any such simulation must rely
on an auxiliary assumption regarding the distribution of h. Given the distribution of h, one can calculate the fraction of admissible environments in which rational ignoring occurs, which provides a simulated frequency of rational ignoring, Pr[F(h) > 0].
The simulation reported here draws realizations of f1, gL and gR under the assumption that they are
independent and uniformly distributed on the unit interval. For f0, the admissibility condition, f1 > f0,
must be satisfied. Therefore, f0 is generated uniformly on the support [0, f1]. After 1,000,000 draws
from h, we find that the inequality F(h) > 0 holds in 61% of the draws. Thus, slightly more than three
4
There is a non-trivial question of units of time. Does the decision maker decide his or her seatbelting action for the entire year?
For each trip? For each mile? Or for each minute in the car? The assumption of daily seatbelt decisions strikes a balance between
annual and continuous time. The odds ratios in the ignoring index do depend crucially on time normalization.
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N. Berg, U. Hoffrage / Journal of Economic Psychology 29 (2008) 792–809
out of every five two-state, two-action environments (drawn from uniformly distributed h) exhibit rational ignoring.
4.6. Paired comparison
Paired comparison is a frequently studied task in the experimental psychology literature. Given N
objects (e.g., city populations, the salaries of a particular group of scientists, or a pool of loan applicants), subjects are presented with randomly drawn pairs (from N-choose-2 possibilities, i.e.,
N(N 1)/2) and asked to rank them (e.g., say which is larger or better among the two). See Gigerenzer,
Hoffrage, and Kleinbölting (1991) and Goldstein and Gigerenzer (2002) for detailed examples. Because
ordering of objects in the presentation of pairs is randomized, the event that the correct answer is
listed first has an experimentally fixed probability of 1/2.
Given unconditional state probabilities of 1/2, one may investigate parameterizations in which rational ignoring is predicted to occur. To simplify, we assume that the two realized signals (values of X)
condition the event x = L symmetrically
f1 ¼ 1=2 þ r;
and
f 0 ¼ 1=2 r;
ð17Þ
where r, 0 < r < 1/2, parameterizes the distance between conditional and unconditional probabilities,
and is referred to as the strength of the signal X.
Table 1 presents a range of parameterizations in which r and gR/gL both vary. When marginal payoffs are highly asymmetric (e.g., when gR/gL = 0.01 or 10), most signals, over a wide range of strengths,
are ignored except for the very strongest (r = .45). On the other hand when marginal gains are identical (gR/gL = 1) or nearly so, ignoring rarely occurs, and only does so for the very weakest (low-r) signals. This result closely follows the intuition and formal results in Delquié (2006) showing a general
Table 1
Rational ignoring as a function of the marginal payoff ratio gR/gL and strength of signal r
gR/gL
0.10
0.50
0.90
1.00
2.00
5.00
10.00
a
r
f1/(1 f1)
f0/(1 f0)
F(h)a
ignoring?
0.01
0.10
0.25
0.45
0.01
0.10
0.25
0.45
0.01
0.10
0.25
0.45
0.01
0.10
0.25
0.45
0.01
0.10
0.25
0.45
0.01
0.10
0.25
0.45
0.01
0.10
0.25
0.45
1.04
1.50
3.00
19.00
1.04
1.50
3.00
19.00
1.04
1.50
3.00
19.00
1.04
1.50
3.00
19.00
1.04
1.50
3.00
19.00
1.04
1.50
3.00
19.00
1.04
1.50
3.00
19.00
0.96
0.67
0.33
0.05
0.96
0.67
0.33
0.05
0.96
0.67
0.33
0.05
0.96
0.67
0.33
0.05
0.96
0.67
0.33
0.05
0.96
0.67
0.33
0.05
0.96
0.67
0.33
0.05
0.81
0.79
0.68
0.90
0.25
0.17
0.42
8.28
0.01
0.14
1.19
15.34
0.00
0.17
1.33
17.05
1.00
0.67
1.67
33.11
15.99
15.17
9.33
69.26
80.98
79.33
67.67
89.53
yes
yes
yes
no
yes
yes
no
no
yes
no
no
no
no
no
no
no
yes
yes
no
no
yes
yes
yes
no
yes
yes
yes
no
Only the sign of F(h) is needed to determine whether ignoring occurs.
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N. Berg, U. Hoffrage / Journal of Economic Psychology 29 (2008) 792–809
inverse relation between the value of information and strength of preference. Following Delquié, the
case of ggR ¼ 1 would correspond to prior indifference regarding which action is better, which is preL
cisely when the signal X becomes so valuable that it can never be ignored. On the other hand, one
can interpret cases where ggR is very large or small as indicating a strong prior preference for choosing
L
one way or the other, which results in a low value of information and high likelihood of rational
ignoring.
This finding makes a prediction about laboratory studies involving paired choice in which subjects
have the chance to pay attention to or ignore information. If ignoring is to be understood as consistent
with expected-payoff maximization, one would need to demonstrate that the minimum strength of
cues at which those cues begin to be ignored is negatively correlated with asymmetry of the marginal
payoffs. In the paired comparison task, asymmetric marginal payoffs seem unlikely, because one
would need a theory of why subjects have strict preferences over winning one dollar for correctly
ranking items listed in ascending order over winning one dollar for correctly ranking items listed in
descending order. Rational ignoring as modeled above does not therefore appear to provide a promising explanation for the empirical phenomena reviewed in the first two sections of this paper unless
payoff asymmetry can be established. The results of this section imply the more payoff asymmetry,
the broader is the range of signals that can be rationally ignored. Asymmetry may arise, however, even
when monetary payoffs are completely symmetric. For example, if subjects receive greater psychic
payoff from correctly answering challenging questions than for easy ones, then total payoffs (after
aggregating monetary and psychic components) will be asymmetric when experimenters pay a constant quantity of experimental currency units for each correct answer.
Two frequently cited experiments in which prior probabilities are ignored are Kahneman and Tversky’s (1973) engineer–lawyer problem and Tversky and Kahneman’s (1982) taxi cab problem. In both
studies, respondents were asked to produce posterior probabilities given experimenter-controlled
prior probabilities, hit rates, and false positive rates. These studies showed a surprising lack of sensitivity to changes in base rates, known as base-rate neglect or the base-rate fallacy. To the extent that
experimental subjects perceive asymmetric payoffs (e.g., guessing x = 1 correctly is better than correctly guessing x = 0), it is possible that base-rate neglect is consistent with expected-payoff maximization. It may therefore be worthwhile to investigate changes in behavior resulting from treatments in
which signals of varying strength are tested, just as it would be interesting to collect post-experiment
survey data that could reveal subjective asymmetry in the evaluation of equal monetary payoffs
depending on contextual variables like item difficulty.
5. Environments where action is a continuous variable
This section provides an existence proof of the rational ignoring set for the case where the decision
maker’s choice set is a continuum. The continuous action case reveals an intuitive cancellation principle that illuminates how the structure of information and payoffs in the environment interact to produce rational ignoring. Comparing the continuous and discrete-choice cases analyzed in the previous
section, a key difference emerges. Although the rational ignoring set continues to exist in the continuous case, it does so only on a measure-zero subset of the model’s parameter space, in contrast with
the large rational ignoring sets encountered in the discrete-choice case. Psychological and cognitive
realism suggests that human decision makers discretize continuum choice sets, as has been analyzed
in the just-noticeable-differences literature. We introduce an example below providing detail into
how a small amount of discretization in perceptible units of action (or in payoffs or probabilities) leads
once again to positive-measure rational ignoring sets, which can be quite large. Further examples
introduce continuous cues and make concrete the reality that ordinary, non-pathological stochastic
payoff structures are, under certain conditions, consistent with rationally ignoring cues that significantly condition future payoffs.
5.1. Cancellation principle
Consider a family of environments parameterized by h, with K cues that non-trivially enter both the
payoff and conditional density functions
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N. Berg, U. Hoffrage / Journal of Economic Psychology 29 (2008) 792–809
E ¼ fffxjX ðx; X; a; hÞ; pðx; X; a; hÞgjh 2 Hg;
803
ð18Þ
where H represents the universe of admissible values of h, assumed to be non-empty and nonsingleton.
To guarantee a well-behaved expected-payoff objective function (one for which a unique maximizer in a exists, with straightforward first-order conditions characterized by calculus derivatives that
can be moved inside and outside the expectations operator), regularity conditions on p and fxjX are assumed to hold (see, for example, Billingsley, 1995):
A4 (Regularity). For every value of X, there exists a unique maximizer of E[p(x, X, a)jX] with respect to a on
the interior of A, respecting all conditions required for the Theorem of the Maximum and the Implicit
Function Theorem.
Rather than defending this assumption as realistic or general, its restrictiveness serves to strengthen the argument. Showing that rational ignoring can occur even under the most favorable conditions
for constrained optimization implies that the rational ignoring result does not depend on obscure
technical problems involving computation of extrema. Even under conditions ideal for optimization,
unboundedly rational expected-payoff maximizers sometimes ignore relevant predictors. Incorrect
beliefs about probabilities and other cognitive limitations, as well as challenging technical issues in
the solution of the constrained optimization problem (which the assumptions above rule out), only
make it more likely that decision makers might reasonably ignore information.
Provided fxjX(x, X, a) and p(x, X, a) respect assumptions A1–A4, the expected-payoff-maximizing
decision rule ignores relevant information if and only if there exists a cue Xk, such that the function
¼ 0, is constant with respect to Xk over its entire support. In case
a*(x), implicitly defined by oE½pðx;X;aÞjX
oa
fxjX(x, X, a) and p(x, X, a) are differentiable with respect to Xk, a necessary and sufficient condition for
rational ignoring is
Z " 2
o pðx; X; aÞ
opðx; X; aÞ ofxjX ðx; X; aÞ opðx; X; aÞ ofxjX ðx; X; aÞ
fxjX ðx; X; aÞ þ
þ
oX k
oa
oaoX k
oa
oX k
X
#
o2 fxjX ðx; X; aÞ
dX ¼ 0:
ð19Þ
þpðx; X; aÞ
oaoX k
The proof is implicit differentiation of oE½pðx;X;aÞjX
¼ 0 with respect to Xk.
oa
Eq. (19) provides a complete characterization of the set of ignoring environments with continuous
action space and continuously valued cues satisfying A1–A4. Thus, the ignoring set I is defined as
I ¼ fffxjX ; pg 2 E
j
Eq: ð19Þ holdsg:
ð20Þ
A strict subset of I is the set of environments in which cancellation of payoffs and probabilities occurs
C ¼ fffxjX ; pg 2 I
j
pfxjX
is independent of X k
for some k ¼ 1; . . . ; Kg:
ð21Þ
The set C reveals an intuition about inversely related probabilities and payoffs that cancel under the
expectations operator and lead to rational ignoring.
An obvious limitation of the cancellation set C and ignoring set I is that they both occur on a measure-zero subset of E. The virtue of C, however, is that its members possess a recognizable feature of
the decision-making environment and therefore offer hope that rational ignoring environments might
be identified in the laboratory and the field. The next example illustrates the cancellation principle and
shows how just-noticeable differences facilitate the re-emergence of a rational ignoring set with
strictly positive measure in E.
5.2. Berry gathering with diminishing nutritional returns
To highlight the importance of continuous versus discrete actions (rather than continuous versus discrete cues or states of nature), this example features a continuously valued action variable a chosen from
the non-negative real line, while maintaining the binary signals and binary states of nature from previous models in Section 4. To fix ideas, suppose signals and states of nature are interpreted as variables
indicating large versus small harvests from strawberry and raspberry gathering, respectively. Suppose
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that strawberries share with raspberries a common set of weather conditions and precede raspberries in
the gathering season. Therefore, the quantity of strawberries harvested is observed first and provides a
signal about the states of nature defined by the sizes of raspberry harvests later in the season.
The signal X = 1 represents large strawberry harvests, and X = 0 represents small strawberry harvests. This cue provides a statistically valid signal for forecasting raspberry harvests, represented by
x = L for large raspberry harvests and x = R for small raspberry harvests. Assume that a large strawberry harvest implies a greater than 50% chance of a large raspberry harvest, and a small strawberry
harvest implies a lower than 50% chance of a large raspberry harvest:
Prðx ¼ LjX ¼ 1Þ f1 > 1=2;
Prðx ¼ LjX ¼ 0Þ f0 < 1=2:
ð22Þ
ð23Þ
The decision variable a, a P 0, represents the amount of time allocated to raspberry gathering.5 One
naturally conjectures that the decision maker should choose higher values of a whenever X (which is
positively correlated with x) is observed to be high. However, larger than average strawberry harvests
earlier in the season leave berry gatherers more nutritionally fortified, with extra stored energy and, consequently, reduced marginal payoffs from additional berry consumption. This is formalized with the following stylized payoff function:
pðx; X; aÞ ¼ a logð1 þ X þ a1ðx ¼ LÞÞ ca;
ð24Þ
where a > 0 scales the marginal nutritional benefit of berry consumption, c > 0 is the marginal cost of
time, and 1() is an indicator function returning 1 when its argument is a true statement. According to
the functional specification above, the berry gatherer’s season-long accumulation of nutrition is represented by the sum of an initial allocation of 1, the cue value X which contributes 0 or 1 depending on
the size of the strawberry harvest, and a1 (x = L) from raspberry consumption. The term a1(x = L) is
equal to a (the time allocated to raspberry gathering) if the large-harvest state of nature prevails
and zero otherwise. The sum representing accumulated nutrition is transformed by the concave natural logarithm function to reflect diminishing marginal payoffs from additional berry consumption.
The expected-payoff conditional on X is
f1 a logð2 þ aÞ þ ð1 f1 Þa logð2Þ ca if X ¼ 1;
ð25Þ
E½pðx; X; aÞjX ¼
if X ¼ 0:
f0 a logð1 þ aÞ ca
Assuming the parameters satisfy the following admissibility conditions that guarantee an interior
maximizer a* > 0 (away from the boundary a = 0)
f1 > 2c=a
and f 0 > c=a;
it is straightforward to show that the expected-payoff-maximizing action is
2 þ af1 =c if X ¼ 1;
a ðXÞ ¼
1 þ af0 =c if X ¼ 0:
ð26Þ
ð27Þ
The relevant condition for rational ignoring identifies cases where a* is independent of X, which is satisfied whenever a*(1) = a*(0), that is, whenever:
f1 f0 ¼ c=a:
ð28Þ
Thus, ignoring occurs when the informational benefit f1 f0 (i.e., change in likelihood of large raspberry harvests with respect to different observed levels of strawberry harvests) is exactly offset by
the benefit-scaled cost of effort c/a that is required to make use of the information provided by the
cue. We emphasize that rational ignoring occurs in this example without any cognitive bounds or
informational limitations. However, in contrast with previous models in which the action variable
was discrete, the rational ignoring set in this model is small. Because a is continuously valued, the
ignoring set has measure zero in the universe of admissible values of h = [f0, f1, a, c].
5
The decision maker’s information about raspberry harvests is gathered by paying attention to strawberry harvests earlier in the
growing season, whereas the action variable a represents the amount of time allocated to raspberry gathering – two distinct senses
of ‘‘gathering.”
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805
5.3. Berry gathering example continued with just-noticeable differences
Now suppose that time is subjectively perceived in discrete units such that, if a0 is the current
choice of a, then nearby values within > 0 units of a0 are regarded as equivalent
a 2 ða0 ; a0 þ Þ ) a is equivalent to a0 ;
ð29Þ
where represents the minimum change in action that is perceptibly different from the current action,
and the -interval about a0 is an equivalence class. Therefore, if change in some component of X never
moves a*(X) more than , then changes in that signal will imply no perceptible change in action and,
consequently, the signal can be ignored. We investigate the size of the ignoring set relative to the
admissible parameter space as a function of the perceptual limit .
For the case of just-noticeable differences in a with perceptual limit applied to the berry gathering
example, the following inequality represents the condition for rational ignoring: Ignore X if
ja*(X = 1) a*(X = 0)j < . Using the formula for a*(X) from (27), rational ignoring occurs whenever
the exogenous parameters fall within the bounds:
a
ð30Þ
ðf1 f0 Þ 1 < ;
c
or equivalently
c
c
f1 ð1 þ Þ < f0 < f1 ð1 Þ:
a
a
ð31Þ
Inequality (31) defines the rational ignoring set as a subspace comprised of pairs (f1, f0) analogous
f1
f0
;
to Fig. 1, which plotted the rational ignoring set in coordinate pairs of the form 1f
for the case
1f
1
0
of discrete action. The berry gathering model’s admissible parameter space is defined by the four
inequalities: f1 6 1, f0 < f1, 2c/ a < f1, and c/a < f0.
( f0 )
G
F 1- (1- ) c/α
Inadmissible *
because f0 >f1
Inadmissible
because
f1<2c /α
E 1- c/α
D 1- (1+ ) c/α
f0=f1
** Set
Rational Ignoring
H
I
c/α
A
2 c /α
a* conditions on X
B
Inadmissible
because f0<c/α
C
( f1 )
Fig. 2. Rational ignoring with continuous action and just noticeable differences. *The admissible parameter space is given by the
trapezoid ACGH, and the rational ignoring set is given by segment AE whose equation is given by: f0 = f1 c/a. **Just noticeable
differences makes an equivalence class of all values within e units of the current level of a, and the rational ignoring set expands
around the zero-measure segment AE to the positive-measure polygon ABDEFI.
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N. Berg, U. Hoffrage / Journal of Economic Psychology 29 (2008) 792–809
Fig. 2 depicts the model’s admissible parameter space as the trapezoid ACGH, and the measure-zero
rational ignoring set as the segment AE. With -noticeable-differences in a, the rational ignoring set
grows to the shaded polygon ABDFI. The following is an analytic expression for the ratio of the area
of ABDFI to the area of ACGH can be computed, which gives the rate of rational ignoring r (under
the assumption of uniformly distributed f1 and f0):
rðÞ ¼ ð4c=aÞ½1 ðc=aÞ=ð1 2c=aÞ;
ð32Þ
for 0 6 < 1 and c/a < 1/2. Thus, the rate of rational ignoring is approximately proportional to . Double
the coarseness and the size of the rational ignoring set roughly doubles as well. Numerical simulations
allow one to explore this parameter space further under different distributional assumptions, and one
consistently finds that just-noticeable differences grow the ignoring set to sizable measurable proportions even for modest coarseness of the perceived action space or discretizing social conventions (e.g.,
telling time to the nearest quarter of an hour, etc.).
5.4. Example with continuous outcomes and cues
This section provides two final examples of the continuous action model – this time with continuous outcomes and cues. This helps illustrate how ordinary and non-pathological functional forms can
give rise to rational ignoring.
Suppose the joint pdf of x and X is
fxX ðx; X; aÞ ¼ x þ X
for x;
X 2 ½0; 1;
and 0 otherwise:
ð33Þ
Computing the marginal density fX and the ratio fxX/fX, it is easy to show that the conditional pdf of x
given X is
fxjX ¼
xþX
1
þX
2
for x;
X 2 ½0; 1;
and 0 otherwise:
ð34Þ
The cue is relevant because the conditional expectation of x given X depends non-trivially on X
Z 1
xþX
1
1 X
dx ¼ 1
E½xjX ¼
x1
þ
; X 2 ½0; 1;
ð35Þ
þX
þX 3 2
0
2
2
which obviously has a non-zero derivative with respect to X.
To illustrate the cancellation principle, suppose the payoff function p is given by the product of the
reciprocal of fxjX and a simple function of a with a global maximum
pðx; X; aÞ ¼
1
2
þX
ða a2 Þ:
xþX
ð36Þ
The expected payoff simplifies to
E½pðx; X; aÞjX ¼ a a2 ;
ð37Þ
which is independent of X, and a ¼ 12 for every possible observed value of X, despite the fact that both
fxjXandp depend non-trivially on X.
5.5. Ignoring one of two continuous cues
In the previous continuous-cue environment, the conditional pdf of x was independent of action a;
the payoff function was quasi-concave in a, building in a well-defined interior maximum; and the
optimal decision rule was constant. The following example demonstrates that none of these features
is necessary for the ignoring condition (19) to hold.
Consider an extension of the previous example, this time with two cues, X1 and X2, and joint pdf:
fx;X 1 ;X 2 ðx; X 1 ; X 2 Þ ¼ aðx þ 2X 1 X 2 Þ;
X 1 2 ½0; 1; X 2 2 ½0; 1; x 2 ½0; a;
ð38Þ
1=2
where the upper bound on x is the constant a 12 þ 2a þ 14
, and a > 0. Integrating out x, the joint
pdf of X1 and X2 (on its support) is
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N. Berg, U. Hoffrage / Journal of Economic Psychology 29 (2008) 792–809
fX 1 ;X 2 ðX 1 ; X 2 Þ ¼
Z
a
0
1 2
aðx þ 2X 1 X 2 Þdx ¼ a
a þ 2aX 1 X 2 ;
2
807
ð39Þ
and the conditional pdf of x with respect to (X1, X2) (on its support) is
x þ 2X 1 X 2
;
a2 þ 2aX 1 X 2
2
fxjX 1 ;X 2 ðx; X 1 ; X 2 Þ ¼ 1
ð40Þ
which depends on a non-trivially through a.
Now consider the payoff function:
1
pðx; X 1 ; X 2 Þ ¼ 2
a2 þ 2aX 1 X 2
ðX 2 2xÞ:
x þ 2X 1 X 2
ð41Þ
In contrast to the previous example, the payoff function here is monotonic in a with no built-in interior maximum. The expected payoff conditional on X1 and X2 is
Z a
Z a
pðx; X 1 ; X 2 ÞfxjX 1 ;X 2 ðx; X 1 ; X 2 Þdx ¼
ðX 2 2xÞdx ¼ X 2 a a2 :
ð42Þ
0
0
The condition a = X2/2 implicitly defines the optimal action
a ¼ 8=ðX 22 þ 2X 2 Þ:
ð43Þ
The non-constant decision rule a* ignores the cue X1 while making use of X2, even though both cues are
payoff relevant.
6. Conclusion
Ignoring is, in many cases, manifestly detrimental to the well being of decision makers and the
societies they populate. One need only think of global warming as an example in which failing to
pay attention to initially subtle signals of future outcomes can have drastic consequences. The ubiquity of ignoring, even when the stakes are very high, prompts the question of why humans, who can
be exceedingly sensitive to subtle signals in some contexts, systematically ignore predictive information. This paper demonstrates that throwing away information is, contrary to intuition, consistent with expected utility maximization. Nothing pathological is required. It requires only
sufficient asymmetry in payoffs to rationalize ignoring, consistent with recent developments showing that the value of information falls as prior preferences become more intense or decisive (Delquié, 2006).
This paper avoids making categorical claims about the benefits of ignoring. From a descriptive
point of view, it shows that ignoring is not always detrimental, and that it is wrong to automatically
label decision makers who throw away information as irrational. Expected-payoff-maximizing decision rules in a variety of stochastic payoff environments are shown to ignore information even
though that information unambiguously helps predict payoff-relevant events in the future. The normative implications of ignoring are therefore ambiguous and must be analyzed on a case-by-case or
context-specific basis. Rather than automatically concluding that subjects are irrational, experimental studies reporting evidence of ignoring would benefit from analysis of the match between those
decision processes that led to ignoring and the contexts in which they are used. This would require
analyzing the extent to which ignoring provides a shortcut to achieving high levels of performance,
one marker of which is negatively correlated marginal effects of the signal on payoffs and
probabilities.
This paper attempts to go as far as possible in studying ignoring without introducing decision costs,
cognitive limitations or other forms of bounded rationality. The motivation for this is to isolate structure in the stochastic payoff environment that is friendly to decision strategies which ignore information. Thus, the theory of ecological or environment-based ignoring is both distinct from, and
complementary to, bounded rationality, suggesting an enlarged set of joint explanations for the success of information-frugal decision rules.
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N. Berg, U. Hoffrage / Journal of Economic Psychology 29 (2008) 792–809
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